New coprime vertex labelings

A coprime vertex labeling is an injective assignment of the labels {1, 2, . . . , n} to the vertices of an n-vertex simple connected graph such that adjacent vertices receive relatively prime labels. I will present new labelings for several infinite families of graphs. No prior knowledge of graph theory will be assumed. Joint work with Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White, and Alyssa Whittemore.

This talk was given by my undergraduate research student Hannah Prawzinsky at the 2016 Nebraska Conference for Undergraduate Women in Mathematics at the University of Nebraska-Lincoln.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Transcript

1. New Coprime Vertex Labelings Hannah Prawzinsky Joint work with: Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White & Alyssa Whittemore NCUWM January 30, 2016
2. What is a graph? Definition A graph G(V, E) is a set V of vertices and a set E of edges connecting some (possibly empty) subset of those vertices.
3. Simple graphs Definition A simple graph is a graph that contains neither “loops” nor multiple edges between vertices. For the remainder of the presentation, all graphs are assumed to be simple. Here is a graph that is NOT simple.
4. Connected and unicyclic graphs Definition A connected graph is a graph in which there exists a “path” between every pair of vertices. For the remainder of the presentation, all graphs are assumed to be connected. Definition A unicyclic graph is a simple graph containing exactly one cycle. Here is a unicyclic graph that is NOT connected.
5. Infinite families of graphs P8 C12 S5
6. Graph labelings Definition A graph labeling is an “assignment” of integers (possibly satisfying some conditions) to the vertices, edges, or both. Formal graph labelings are functions. 2 3 2 3 1 4 1 4 1 2 3 4 1 2 3 4
7. Prime vertex labelings Definition An n-vertex graph has a prime vertex labeling if its vertices are labeled with the integers 1, 2, 3, . . . , n such that no label is repeated and all adjacent vertices (i.e., vertices that share an edge) have labels that are relatively prime. 1 6 7 4 9 2 3 10 11 12 5 8 Some useful number theory facts: • All pairs of consecutive integers are relatively prime. • Consecutive odd integers are relatively prime. • A common divisor of two integers is also a divisor of their difference. • The integer 1 is relatively prime to all integers.
8. Original motivation for research Conjecture (Seoud and Youssef, 1999) All unicyclic graphs have a prime vertex labeling. Though our research lead to many new results for unicyclic graphs, some of which were presented last year, this talk will primarily focus on a a specific family of non-unicyclic graphs.
9. Known prime labelings 1 2 3 4 5 6 7 8 P8 1 12 11 10 9 8 7 6 5 4 3 2 C12 1 2 6 5 4 3 S5
10. Cycle Chains Definition A cycle chain, denoted Cm n , is a graph that consists of m different n-cycles adjoined by a single vertex on each cycle (each cycle shares a vertex with its adjacent cycle(s)). Here we show labelings for Cm 4 , Cm 6 , and Cm 8 . The labelings for these three infinite families of graphs all employ similar strategies.
11. Example of C4 8
12. Cycle chain results Theorem All Cm 8 , Cm 6 , Cm 4 have prime vertex labelings.
13. Labeled C5 8 1 2 3 4 5 6 7 8 15 11 10 9 1 12 13 14 19 18 17 16 15 22 21 20 29 25 24 23 19 26 27 28 33 32 31 30 29 36 35 34
14. Labeled C5 6 1 2 3 4 5 6 11 8 7 1 9 10 16 13 12 11 14 15 19 18 17 16 21 20 26 23 22 19 24 25
15. Labeled C4 4 5 4 3 2 7 6 5 8 11 9 7 10 13 12 11 1
16. Labeled C5 4 5 4 3 2 7 6 5 8 11 9 7 10 13 12 11 14 1 15 13 16
17. Mersenne Primes Definition A Mersenne prime is a prime number of the form Mn = 2n − 1. There are 48 known Mersenne primes. The first few Mersenne primes are: M2 = 22 − 1 = 3 M3 = 23 − 1 = 7 M5 = 25 − 1 = 31
18. Mersenne cycle chains Theorem All Cm n , where n = 2k and 2k − 1 is a Mersenne prime, have prime labelings.
19. Fibonacci Chains Fibonacci sequence The sequence, {Fn}, of Fibonacci numbers is defined by the recurrence relation Fn = Fn−1 + Fn−2 , where F1 = 1 and F2 = 1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . Proposition Any two consecutive Fibonacci numbers in the Fibonacci sequence are relatively prime. Theorem Fibonacci Chains, denoted Cn F , are prime for all n ∈ N where n is the number of cycles that make up the Fibonacci chain.
20. Fibonacci Chains (C5 F ) 1 2 4 3 5 6 7 10 9 8 12 11 13 14 15 16 17 18 19 20 21
21. Future Work and Acknowledgements Future Work • Seoud and Youssef’s Conjecture Acknowledgments • NCUWM Organizers • University of Nebraska—Lincoln • Center for Undergraduate Research in Mathematics • Northern Arizona University • Research Advisors Dana Ernst and Jeff Rushall
22. Questions ?
23. A Labeled 7-Hairy Cycle 1 2 3 4 5 6 7 8 19 17 18 20 21 22 23 24 11 9 10 12 13 14 15 16 29 25 26 27 28 30 31 32
24. Prime Vertex Labeling of C5 P2 S6 1 5 2 3 4 6 7 8 9 13 10 11 12 14 15 16 17 19 18 20 21 22 23 24 25 29 26 27 28 30 31 32 33 37 34 35 36 38 39 40
25. Prime Vertex Labeling of Cn P2 S3 S3 1 2 5 9 11 3 4 6 7 8 10 12 13 14 15 16 19 23 25 17 18 20 21 22 24 26 27 28 29 32 31 35 41 30 33 34 36 37 38 39 40 42 43 44 47 51 53 45 46 48 49 50 52 54 55 56
26. A Labeled Bertrand Weed Graph 1 2 13 10 9 11 14 7 12 8 5 4 3 6
27. Example of H8 (Hastings Helms) 5 4 3 2 1 16 7 6 12 11 10 9 8 15 14 13
28. Book Generalizations Here is an example of the prime labeling for Sn × P6 , in particular, S4 × P6 : 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 5 24 25 26 27 28 29 30 6 1 2 3 4 23